Now, consider the last argument, \(\nabla g_{\lambda }\) , in the gradient descent method.
From Theorem REF , observe that computing the gradient \(\nabla g_{\lambda }(\mathbf {w})\)  involves finding the values of the expression of the form \(L_{\mathbf {t}}^{-1} \mathbf {u}\)  twice, first for computing \(\widetilde{\beta }_{\mathbf {t}}\)  (using (REF )) and then for computing the vector \(\mathbf {c}_{\mathbf {t}}\)  (defined in Theorem REF ). Since \(L_{\mathbf {t}}\)  is a \({p \times p}\)  matrix, computing the matrix inversion \(L_{\mathbf {t}}^{-1}\) 
can be problematic particularly in high-dimensional cases where \(p\)  can be very large [1]}. Since \(L_{\mathbf {t}}^{-1} \mathbf {u}\)  is the solution of the linear equation \(L_{\mathbf {t}} \mathbf {z} = \mathbf {u}\) , in Section REF , we propose a conjugate gradient based approach for solving this linear equation efficiently without the need to compute \(L_{\mathbf {t}}^{-1}\) .
